On the smoothness in the weighted Triebel-Lizorkin and Besov spaces via the continuous wavelet transform with rotations (Q6592851)

Wavelet methods are used to establish boundedness of weak solutions of certain constant coefficient differential operators between Triebel-Lizorkin spaces. Set\N\[\N(U(a,b)h)(x)=T_bJ_a h(x)=a^{-n/2} h(a^{-1}(x-b)),\quad a>0,\ b\in\mathbb{R}^n\N\]\Nand define the continuous wavelet transform of \(f\in L^2(\mathbb{R}^n)\) as\N\[\N(L_h f)(a,b)=\langle f, U(a,b) h\rangle =\int_{\mathbb{R}^n} f(x) \overline{(U(a,b)h)(x)}\, dx\N\]\Nwhere \(h\) is an admissible continuous wavelet. One incorporates the rotation operator \((A_R h)(x)=h(R^{-1}x)\) where \(R\in SO(n)\) and defines\N\[\N(L_h f)(a,b,R)=\langle f, U(a,b,R) h\rangle; (U(a,b,R) h)(x)=a^{-n/2} h(a^{-1} R^{-1}(x-b))\N\]\Nwith the admissibility condition of square-integrability of \(U(a,b,R)h\) with respect to \(dR db a^{-(n+1)} da\). Reproducing formulas are reviewed.\N\NA weight function \(k:\mathbb{R}^n\to \mathbb{R}_+\) is temperate if \(k(x+y)\leq (1+M|x|)^N k(y)\) for some fixed \(N,M>0\) and \(\|f\|_{p,k}\) is the corresponding weighted \(L^p\)-norm. For \(1\leq p\), \(q<\infty\), \(r\in (0,1)\) and \(c\in \mathbb{R}^n\setminus \{0\}\) the weighted Triebel-Lizorkin space \(F^{r,q}_{p,k}\) consists of those \(f\in L^{p}_{k}(\mathbb{R}^n)\) such that \(\|\Delta_c f\|_{L^q(\mathbb{R}^n, |c|^{-rq-n})}=\left(\int_{\mathbb{R}^n} |(\Delta_c f)(\cdot)|^q \frac{dc}{|c|^{rq+n}}\right)^{1/q}\in L^p_k\) where \((\Delta_c f)(x)=f(x+c)-f(x)\). It is shown that for \(t\in (0,1)\), \(F^{m+t,q}_{p,k}=\{f\in L^p_k(\mathbb{R}^n):\forall |\beta|\leq m,\ \partial^\beta f\in F_{p,k}^{t,q}(\mathbb{R}^n)\}\). Characterization of \(F^{r,q}_{p,k}\) in terms of the wavelet transform is fundamental to this result. Define \[ R_\partial^{-\alpha} h = \sum_{\alpha=(|\beta|,|\gamma|,\dots,,|\eta)}\frac{\alpha!}{\beta!\gamma!\cdots\eta !} R_1^{-\beta}R_2^{-\gamma}\cdots R_n^{-\eta} \partial^{\beta+\gamma+\dots \eta} h\] and \[ S_\partial^{\gamma}=\sum_{|\beta|\leq m} c_\beta R_\partial^{-\beta+\gamma}\, .\] The main theorem assumes that \(h\) is admissible and satisfies \(\| \|| (\Delta_c f)(\cdot)\||_{q,r} \|_{p,q}=\mathcal{O}(a^r)\) as \(a\to 0\) and \(\|| (\Delta_c S_\partial^{\gamma}h(\cdot)\||_{q,r}\in L^1(\mathbb{R}^n)\) where \(|\gamma|\leq m\). The theorem then states that if \(u\in W^{m,p}(\mathbb{R}^n)\) is a weak solution of \(Qu=f\) where \(Q=\sum_{|\beta|\leq m} c_\beta \partial^\beta\) with \(f\in F_{p,k}^{r,q}(\mathbb{R}^n)\), then for \(p>\frac{2N}{r}\) (\(N\) in the definition of moderate weight \(k\)) one has \(u\in F_{p,q}^{m+r,q}(\mathbb{R}^n)\). Here \(\||\cdot\||_{r,q}\) is an abbreviation for \(\|\cdot\|_{L^q(\mathbb{R}^n, |c|^{-rq-n})}\).\N\NA corresponding result is proved for Besov spaces, which are defined by reversing the inner and outer norms that define the weighted Triebel-Lizorkin spaces, namely \(f\in B_{p,k}^{q,r}(\mathbb{R}^n)\) if \(\|| \|(\Delta_c f)(\cdot)\|_{p,k}\||_{q,r}<\infty\). Under the same conditions on the weight and admissibility function in the norm definitions, if \(u\in W^{m,p}(\mathbb{R}^n)\) is a weak solution of \(Qu=f\) with \(f\in B_{p,k}^{r,q}(\mathbb{R}^n)\) then for \(p>\frac{2N}{r}\) one has \(u\in B_{p,q}^{m+r,q}(\mathbb{R}^n)\). In these results, the ability to absorb the differential operator into a suitably defined admissible wavelet generator accounting for rotations plays a fundamental technical role.